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%I #16 Nov 05 2019 13:55:20
%S 1,1,2,7,31,158,884,5292,33385,219797,1500449,10573815,76688602,
%T 571232869,4363912280,34161879247,273906591562,2248935278231,
%U 18909284838057,162842178607893,1436660527685476,12988076148036405,120345643023918566,1143054910071718088,11129160383826078389
%N G.f. satisfies: A(x) = Sum_{n>=0} x^n * A(x)^(n^2).
%H Paul D. Hanna, <a href="/A107595/b107595.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) = (1/x)*Series_Reversion(x/F(x)) and thus A(x) = F(x*A(x)) where F(x) is the g.f. of A107594.
%F G.f. A(x) = x/Series_Reversion(x*G(x)) and thus A(x) = G(x/A(x)) where G(x) is the g.f. of A107596.
%F From _Paul D. Hanna_, Apr 23 2010: (Start)
%F Let A = g.f. A(x), then A satisfies the continued fraction:
%F A = 1/(1 - A*x/(1 - (A^3-A)*x/(1 - A^5*x/(1 - (A^7-A^3)*x/(1 - A^9*x/(1- (A^11-A^5)*x/(1 - A^13*x/(1 - (A^15-A^7)*x/(1 - ...)))))))))
%F due to an identity of a partial elliptic theta function. (End)
%F From _Paul D. Hanna_, May 04 2010: (Start)
%F Let A = g.f. A(x), then A satisfies:
%F A = Sum_{n>=0} x^n*A^n * Product_{k=1..n} (1 - x*A^(4k-3)) / (1 - x*A^(4k-1))
%F due to a q-series identity. (End)
%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 158*x^5 + 884*x^6 + 5292*x^7 +...
%e Let A = g.f. A(x) then
%e A = 1 + x*A^1 + x^2*A^4 + x^3*A^9 + x^4*A^16 + x^5*A^25 ...
%e = 1 + x*(1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 158*x^5 + 884*x^6 +...)
%e + x^2*(1 + 4*x + 14*x^2 + 56*x^3 + 257*x^4 + 1312*x^5 +...)
%e + x^3*(1 + 9*x + 54*x^2 + 291*x^3 + 1557*x^4 + 8568*x^5 +..)
%e + x^4*(1 + 16*x + 152*x^2 + 1152*x^3 + 7836*x^4 +...)
%e + x^5*(1 + 25*x + 350*x^2 + 3675*x^3 + 32625*x^4 +...)
%e + x^6*(1 + 36*x + 702*x^2 + 9912*x^3 + 114201*x^4 +...) +...
%e = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 158*x^5 + 884*x^6 +...
%t m = 25; A[_] = 0;
%t Do[A[x_] = 1 + Sum[x^k A[x]^(k^2) + O[x]^j, {k, 1, j}], {j, m}];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Nov 05 2019 *)
%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(k=1,n,A=1+sum(j=1,n,x^j*A^(j^2)+x*O(x^n)));polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A176719, A107590, A107594, A107596.
%K eigen,nonn
%O 0,3
%A _Paul D. Hanna_, May 17 2005
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